The Coalition Structure Core is Accessible

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The Coalition Structure Core is Accessible by

László Á. KÓCZY Luc LAUWERS

Econometrics

Center for Economic Studies

Discussions Paper Series (DPS) 02.19

http://www.econ.kuleuven.be/ces/discussionpapers/default.htm December 2002

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L ÁSZL Ó Á. K ÓCZY AND LUC LAUWERS

Abstract. For each outcome (i.e. a payoff vector augmented with a coalition structure) of a TU-game with a non-empty coalition structure core there exists a finite sequence of successively dominating outcomes that terminates in the coalition structure core. In order to obtain this result a restrictive dominance relation - which we call enforceable dominance - is employed.

1. Introduction

For a TU-game in coalitional form, there are two fundamental and strongly linked problems: (i) what coalitions will form, and (ii) how will the members of these coalitions distribute their total worth. We attempt to answer these questions for a certain class of games. We presuppose some bargaining process and show that the coalition structure core, provided it is non-empty, comes forward as a natural candidate for a solution.

We consider a TU-game and some initial individually rational payoff configuration - i.e. a coalition structure augmented with an individual rational and group rational payoff vector (Owen 1982, p236). In case some coalition could gain by acting for themselves, it can reject this initial outcome and propose a second outcome. We impose an enforceability-condition upon such a counter-proposal: first, the deviating coalition is a member of the new coalition structure; second, none of the players in the deviating coalition loose when moving towards the new outcome; third, the new coalition structure also contains those coalitions in the initial configuration that do not shelter deviating players; and fourth, these unaffected coalitions obtain the very same payoffs. Then again, another coalition may reject this counter-proposal in favour of a third outcome, and so forth. Apparently, this bargaining process turns the coalition structure core, if non-empty, into an accessible set of outcomes:

For each outcome of a TU-game with a non-empty coalition structure core, there exists a finite sequence of successive ‘enforceable’ counter-proposals that terminates in the coalition structure core.

Date: October 3, 2001.

Key words and phrases. Coalition structure, core-extension, non-emptiness, dominance.

This paper benefited from presentations at the LGS2 Conference (St Petersburg, June 2001) and at the Warwick Summer Research Conference (July 2001). Discussions with Hideo Konishi and Licun Xue have led to a considerable improvement of the paper. The first author gratefully acknowledges the joint support (SOR-H/99/23 and SOR-H/00/01) of the Katholieke Universiteit Leuven and the Soros Foundation.

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2 L ÁSZL Ó Á. K ÓCZY AND LUC LAUWERS

In the search for a dynamic foundation of the core, already Green (1974) made an im-portant contribution. He established a finite process of successive counter-proposals that almost surely reaches the core. Later on, Wu (1977) showed the existence of a bargaining scheme that converges to the core and rephrased this result as the core is globally stable. Finally, our result is a continuation of the work by Sengupta and Sengupta (1996). Formu-lated in the language of von Neumann and Morgenstern, they proved the indirect stability of the core: no payoff allocation dominates a core outcome, and each outcome is indirectly dominated by a core outcome. We refine this stability property in two dimensions.

First, Sengupta and Sengupta (and also Green and Wu) concentrate on the core. Hence they do not tackle problem (i). They take the coalition structure to be exogenously given and assume that the grand coalition forms. We also consider the coalition formation process and extend the stability result to the coalition structure core.

Second, we employ a more restrictive dominance relation based upon an enforceability condition. Sengupta and Sengupta allow a deviating coalition to affect the payoffs of all the players, and thus ignore the behaviour and the motivation of the outsiders. We impose, as already indicated, strong restrictions upon counter-proposals: the deviating players cannot intervene in the structure and in the payoffs of those coalitions that are not involved when they separate to form a coalition.

The next section collects preliminaries, introduces the coalition structure core, and defines enforceable dominance. Section 3 studies dominating chains and proves our result. The coalition structure core is characterised as the smallest set of outcomes that satisfies this accessibility property.

2. Preliminaries

We introduce the notation and define games, outcomes, dominance, and a core concept. As we do not assume that the grand coalition forms our notion of outcome slightly differs from the usual notion of imputation.

Let N = {1, 2, . . . , n} be a set of n players. Non-empty subsets of N are called coalitions. A partition is a set of pairwise disjoint coalitions so that their union is N and represents the breaking up of the grand coalition N . For a partition P = {C1, C2, . . . , Cm} and a

coalition C, the partners’ set P (C, P) of C in P is defined as the union of those coalitions in P that have a non-empty intersection with C:

P (C, P) = {i |i ∈ Cj with j such that Cj∩ C 6= ∅} =

[

Cj∩ C 6= ∅

Cj.

A characteristic function v : 2N _{\{∅} −→ R assigns a real value to each coalition. The}

pair (N, v) is said to be a transferable utility game in characteristic function form, in short, a game. An outcome of a game (N, v) is a pair (x, P) with x in Rn _{and P a partition of}

N . The vector x = (x1, x2, . . . , xn) lists the payoffs of each player and satisfies

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with x(C) =P

j∈Cxj. The first condition is known as individual rationality: player i will

cooperate to form a coalition only if his payoff xi exceeds the amount he would get on his

own. The second condition combines feasibility and the myopic behaviour of the players, it states that each coalition in the partition P allocates its value among its members. Outcomes with the same payoff vector are said to be payoff equivalent. The set of all outcomes is denoted by Ω(N, v).

In case the grand coalition forms, then an outcome is a pair (x, P) with P = {N }, xi ≥ v({i}), and x(N ) = P_i∈Nxi = v(N ). As such, outcomes generalise imputations.

Also note that Ω(N, v) is non-empty: it contains the outcome in which the grand coalition is split up in singletons.

Now, we define an enforceable dominance relation. An interpretation and a discussion follows. Later on, we will drop the adjective ‘enforceable’, this shall not lead to confusion. Definition 1. Let x, y ∈ Rn and let C be a coalition. Then, vector x dominates y by C, denoted by x >C y, if

- for each player i in C we have xi ≥ yi, and

- for at least one player i in C we have xi > yi.

Let (N, v) be a game and let a = (x, P) and b = (y, Q) be two outcomes. Then, outcome a enforceably dominates b by C if

- P contains C,

- P contains all coalitions in Q that do not intersect C, - the payoff vector x dominates y by C, and

- the restrictions of x and y to the set of players outside P (C, Q) coincide.

Outcome a enforceably dominates b if P contains a coalition C such that a dominates b by C.

This relation should be interpreted in a dynamic way. Let a = (x, P) dominate b = (y, Q) by C. Then, if b is considered as the initial outcome, one can say that coalition C deviates and enforces the new outcome a. Indeed, in order to obtain a higher total payoff, coalition C separates from its partners (and at least one member of C gets strictly better off). The players in P (C, Q) \ C become ex-partners of C. They may reorganise themselves and their payoffs might decrease when moving from b to a. In the worst case, these ex-partners fall apart to singletons. Finally, the outsiders, i.e. the players not in P (C, Q), are left untouched.

Definition 1 also models a merger or a breaking up. In the former case, the deviating coalition is the union of some of the coalitions in the initial partition. In the latter case, an initial coalition is split up into two or more subcoalitions; each subcoalition that is better off in the new outcome can be considered as the deviating coalition.

Sengupta and Sengupta (1996) restrict their attention to the core, i.e. they assume that the grand coalition forms. As a consequence, they employ a dominance relation at the level of payoff vectors. However, if one is also concerned with the coalition formation process, enforceable dominance seems to be a natural and a straightforward extension. On the one hand, the idea of enforceability strongly limits the outlook of a dominating sequence, on

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the other, if outcome b is dominated by a at the level of payoff vectors, then there exists an outcome a0 that enforceably dominates b. Therefore, the enforceability condition does not affect the set of undominated outcomes.

Definition 2. Let (N, v) be a game. The coalition structure core C(N, v) is the set of outcomes no coalition can improve upon.

Equivalently, the pair (x, P) is in the coalition structure core if and only if it satisfies feasibility and coalitional rationality:

- for each coalition C in P we have x(C) ≤ v(C), and - for each coalition S we have x(S) ≥ v(S).

Note that the coalition structure core might contain payoff equivalent outcomes. Also, in case the grand coalition forms, the coalition structure core includes the core. Moreover, the linear programming problem to obtain the coalition structure core is very similar to the one behind the core. As a consequence the well-known Shapley-Bondareva conditions that guarantee a non-empty core extend to the present framework. We will not use these con-ditions. Therefore we state the result without explaining the notion of balanced collection (Owen 1982, Chapter 8).

Let (N, v) be a game. Let v∗ be the maximum of v(P) =P

C∈ P v(C) where P runs over

all partitions of N . The number v∗ is called the value of the game (N, v). The value of a superadditive game is equal to the value of the grand coalition.

Proposition. The coalition structure core C(N, v) of a game (N, v) with value v∗ is non-empty if and only if for each balanced collection S we have

X

S∈ S

δSv(S) ≤ v∗,

where the real numbers δS are the balancing weights. Replace the value v∗ in the above

inequalities with the value v(N ) to obtain the (stronger) conditions for a non-empty core. 3. The coalition structure core is accessible

Consider an initial outcome. If a coalition can obtain a higher payoff, it is allowed to deviate and to propose a second outcome, and so forth. This bargaining process gives rise to a ‘dominating’ sequence. We show that for each outcome there exists a dominating sequence that terminates in the coalition structure core. Let (N, v) be a game and let Ω = Ω(N, v) be the set of all outcomes.

Definition 3. Let a, b ∈ Ω. Outcome a is said to be accessible from b, and we write a ← b (or b → a), if one of the following conditions holds

- a and b are payoff equivalent, or

- a sequentially dominates b, i.e. there exists a positive integer k and a sequence of outcomes

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such that ai dominates ai−1for i = 1, 2, . . . , k. The integer k is said to be the length

of (or the number of steps in) the dominating sequence.

The relation ‘←’ describes a possible succession of transitions from one outcome to another. We are interested in the outcomes that appear at the end of these sequences.

Definition 4. Let ∆ be a set of outcomes. Then, ∆ is accessible from Ω if for each b in Ω there exists an a in ∆ such that a ← b.

Lemma. Let (N, v) be a game with a non-empty coalition structure core. Then, the coalition structure core is accessible.

Proof. Let b0 = (y0, Q0) be an outcome that is not in C(N, v). In case b0 is dominated by

an outcome in C(N, v), the proof is done. In case no outcome in C(N, v) dominates b0, we

look for a dominating sequence that terminates in the core.1 This sequence will be denoted by b0 → b1 → b2 → . . .. As a consequence, coalitions and individual payoffs have a double

subscript the first one of which refers to the position in this dominating sequence.

The proof is divided into several steps. First, we indicate how to detect those players that can be blamed for not being able to go to the core in one step. Then, we define a sequence of suitable deviations and show that the core is reached after a finite number of steps. As a matter of fact, we rule out the case that this dominating sequence has an infinite length.

Step 1. Defining the set of overpaid players.

Interpret b0 = (y0, Q0) as the initial outcome. Let a = (x, P) be a core outcome. A player

i for which y0i > xi is said to be overpaid relative to a. Let O(b0, a) collect these overpaid

players. Since b0 is not dominated by a, the set O(b0, a) is non-empty.

Now, we consider the collection of core outcomes that minimise the number of overpaid players. Within this collection, we look for an outcome a∗ = (x∗, P∗) that minimises the amount overpaid y0(O0) − x∗(O0), where O0 = O(b0, a∗). We consider a∗ as a core outcome

close to b0.

Step 2. Selecting a deviating coalition.

Since the outcome b0 = (y0, Q0) is not in the core there exists at least one blocking coalition,

i.e. a coalition D for which v(D) > y0(D). We select a deviating coalition D as follows.

First, we inspect the coalitions in the partition P∗ and we look for a blocking coalition D among P∗. Next, if the partition P∗ does not contain a blocking coalition, then the outcome b0 satisfies y0(N ) = v∗ and is said to be efficient (with respect to P∗), v∗ is the

value of the game. In that case we select a minimal (for inclusion) blocking coalition. Step 3. Defining a deviating outcome.

In order to define the payoff vector in the deviating outcome b1 = (y1, Q1) we consider the

different types of players separately.

First, we deal with the deviating players. Since D blocks b0 and a∗ is a core outcome, we 1_{In this proof we use the term ‘core’ as a shorthand for ‘coalition structure core’.}

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know that y0(D) < v(D) ≤ x∗(D). Let i ∈ D. The payoff y1i depends upon whether or

not D contains overpaid players.

(1) If D does not contain overpaid players, then we define y1i = y0i+ δi(D) ≤ x∗i,

with δi(D) non-negative and adding up to δ(D) = v(D) − y0(D).

(2) If D does contain overpaid players, then we define y1i=

(

y0i+ |D ∩ O1 0|[v(D) − y(D)] in case i is overpaid,

y0i in case i is not overpaid.

In words, the deviating coalition divides the surplus v(D) − y0(D) among its members.

The overpaid players are served first and consume the whole surplus. The non-overpaid players experience either a status quo or an improvement.

Secondly, the ex-partners of D are assumed to split up into singletons. Hence, each player i in P (D, Q0)\D receives his value v({i}) as payoff.2

Thirdly, the outsiders remain untouched: if i /∈ P (D, Q0), then y1i = y0i.

In conclusion: when moving from b0 to b1, the overpaid ex-partners of D become

non-overpaid. In case b1 is either a core outcome or dominated by a core outcome, the proof

finishes. Otherwise, execute the next steps. Step 4. An iteration.

We denote the set O(b1, a∗) of overpaid players in the outcome b1 by O1. This set O1

is a subset of O0. We repeat Steps 2 and 3 and we generate a dominating sequence

b0 → b1 → b2 → . . . of outcomes and a corresponding sequence O0 ⊇ O1 ⊇ O2 ⊇ . . . of sets

of overpaid players. If the core is reached after a finite number of steps, then the iteration stops and the proof is done. Otherwise, we end up with an infinite dominating sequence { bk}k=0,1,2,... each outcome bk being outside the core and not dominated by the core.

Step 5. Inspecting the infinite sequence.

As the core is not reached the set of overpaid players finds its non-empty minimal form after a finite number of steps. Denote this set by O. The iteration procedure is unable to reduce O. Due to the selection criteria for the deviating coalition (Step 2) there is somewhere in the sequence an outcome bt= (yt, Pt) that is efficient with respect to P∗ and

satisfies Ot= O. Let us replace bt with b = (y = yt, P∗).

We claim that the outcome b is in the core. Denote the partners’ set of the overpaid players in P∗ by B, i.e. B = P (O, P∗), and the complement of B by A. Since A does not contain overpaid players and b is efficient with respect to P∗, the payoff vectors y and x∗ restricted to A coincide: y|_A = x∗|_A.

We complete the proof of the claim by contradiction. Let D be a blocking coalition. Obviously, D is not a subset of A. Hence D intersects B. Let ¯D = P (D, P∗) be the partners’ set of D. The efficiency of b implies that y( ¯D) = v( ¯D). Since O can not be

2_{This assumption can be relaxed. The ex-partners are allowed to reorganise themselves provided none}

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reduced, the coalition D contains all the overpaid players in ¯D. Therefore, ¯D \ D only contains non-overpaid players and satisfies y( ¯D \ D) ≤ x∗( ¯D \ D). Use the efficiency of the outcome b together with the fact that a∗ is a core outcome to conclude that y(D) ≥

x∗(D) ≥ v(D). Hence, D is not blocking. A contradiction.

In order to stress the impact of the particular construction in the above proof we give an example of a bargaining scheme that does not enter the coalition structure core. Example. Let Γα = ({1, 2, 3} , v) denote the three-player game, where each singleton

has value 0, each pair has value 2, and v(N ) = α. Let α = 6. The core is non-empty. Nevertheless, the next three outcomes generate a cycle of dominating outcomes:

((1, 1, 0), {1, 2} , {3}) , ((1, 0, 1), {1, 3} , {2}) , and ((0, 1, 1), {2, 3} , {1}) . We close this section with a characterisation of the coalition structure core.

Theorem. The coalition structure core of a game, if non-empty, is the smallest (for inclusion) set of outcomes that satisfies accessibility.

Proof. Accessibility follows from the previous lemma. Furthermore, each outcome in the

coalition structure core is undominated. This implies minimality.

References

Green JR (1974) The stability of Edgeworth’s recontracting process. Econometrica 42(1):21–34 Owen G (1982) Game Theory. Academic Press, New York

Sengupta A, Sengupta K (1996) A property of the core. Games and Economic Behavior 12:266–273 Wu LSY (1977) A dynamic theory for the class of games with nonempty cores. SIAM Journal of Applied

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Copyright © 2002 @ the author(s). Discussion papers are in draft form. This discussion paper is distributed for purposes of comment and discussion only. It may not be reproduced without permission of the copyright holder. Copies of working papers are available from the author.